Problem: If $x+\frac{1}{x}=6$, then what is the value of $x^{2}+\frac{1}{x^{2}}$?
Explanation: Squaring the equation provided, we get $x^2+2(x)\left(\frac{1}{x}\right) +\frac{1}{x^2}=36,$ so $x^2+\frac{1}{x^2}=\boxed{34}.$